Walgreens Boots Alliance Inc. has made a takeover approach to drug distributor AmerisourceBergen Corp., a move that could help boost profitability at the drugstore giant and insulate it against external threats in an increasingly competitive health-care landscape.

Representatives of Walgreens Chief Executive Stefano Pessina reached out several weeks ago to representatives of Amerisource CEO Steven Collis, according to people familiar with the matter. They discussed the possibility of Walgreens buying the portion of Amerisource...